This workshop will highlight the work of several prominent women working in harmonic analysis, including some of the field's rising stars. There will also be a panel discussion. There will also be a contributed poster session. This workshop is open to, and poster contributions are welcome from all mathematicians.

This workshop will highlight the work of several prominent women working in harmonic analysis, including some of the field's rising stars. There will also be a panel discussion. There will also be a contributed poster session. This workshop is open to, and poster contributions are welcome from all mathematicians.

To apply for funding, you must register by the funding application deadline displayed above.

Students, recent Ph.D.'s, women, and members of underrepresented minorities are particularly encouraged to apply. Funding awards are typically made 6 weeks before the workshop begins. Requests received after the funding deadline are considered only if additional funds become available.

MSRI has preferred rates at the Hotel Durant, depending on room availability. Reservations may be made by calling 1-800-238-7268 OR directly on their website. When making reservations, guests must request the MSRI preferred rate. Click on "Promo/Corporate Code" at the top of the page and enter the code 123MSRI (this code is not case sensitive).

MSRI has a preferred rate at the Hotel Shattuck Plaza, depending on room availability. Guests can call the hotel's main line at 510-845-7300 and ask for the MSRI- Mathematical Science Research Inst. discount. To book online visit this page. Click on "Promo/Corporate Code" at the top of the page and enter the code MSRI (this code is not case sensitive).

MSRI has a preferred rates at Easton Hall and Gibbs Hall, depending on room availability. Guests can call the Reservations line at 510-204-0732 and ask for the MSRI- Mathematical Science Research Inst. rate. To book online visit this page, select "Request a Reservation" choose the dates you would like to stay and enter the code MSRI (this code is not case sensitive).

When does a given set contain a copy of your favourite pattern (for example, specially arranged points on a line or a spiral, or the vertices of a polyhedron)? Does the answer depend on how thin the set is in some quantifiable sense? Problems involving identification of prescribed configurations under varying interpretations of size have been vigorously pursued both in the discrete and continuous setting, often with spectacular results that run contrary to intuition. Yet many deceptively simple questions remain open. I will survey the literature in this area, emphasizing some of the landmark results that focus on different aspects of the problem

We discuss historic aspects of 'sharp weighted estimates', notably its origins in simple probability spaces with dyadic filtrations. Some cornerstone results central to harmonic analysis will be mentioned in a historic fashion. The main part of the lecture returns to the probabilistic setting and stochastic analysis in filtered spaces with continuous time, arbitrary underlying measure and possible 'jumps'. It is proved - via two important methods from 'weights', Bellman and Sparse, that 'all' subordinate pairs of martingales have a sharp A_p bound. It is remarkable that in this generality, not even the boundedness of the maximal function seems to have been known - a fact that presents no difficulty at all with today's methods. This lecture should also serve as an introduction to techniques in 'weights' for those participating in the semester

In a foundational paper, Coifman, Rochberg and Weiss characterize the norm of the commutator [b, T] - where T is a Calderon-Zygmund operator - in terms of the BMO norm of the symbol function b. In this talk, we discuss a two-weight version of this result. Such a result was first obtained by Bloom in 1985, in the one-dimensional case, for the Hilbert transform. More recently, this was extended to the n-dimensional case, for all CZOs, using the modern methods of dyadic harmonic analysis

A very captivating question in solid state physics is to determine/understand the hierarchical structure of spectral features of operators describing 2D Bloch electrons in perpendicular magnetic fields, as related to the continued fraction expansion of the magnetic flux. In particular, the hierarchical behavior of the eigenfunctions of the almost Mathieu operators, despite significant numerical studies and even a discovery of Bethe Ansatz solutions has remained an important open challenge even at the physics level.

I will present a complete solution of this problem in the exponential sense throughout the entire localization regime. Namely, I will describe, with very high precision, the continued fraction driven hierarchy of local maxima, and a universal (also continued fraction expansion dependent) function that determines local behavior of all eigenfunctions around each maximum, thus giving a complete and precise description of the hierarchical structure. In the regime of Diophantine frequencies and phase resonances there is another universal function that governs the behavior around the local maxima, and a reflective-hierarchical structure of those, a phenomena not even described in the physics literature.

These results lead also to the proof of sharp arithmetic transitions between pure point and singular continuous spectrum, in both frequency and phase, as conjectured since 1994. The talk is based on papers joint with W. Liu

In this talk, we consider non-homogeneous, second order, uniformly elliptic systems of partial differential equations. We show that, within a suitable framework, we can define the fundamental solution and the Green functions on arbitrary open subsets. Moreover, we can prove uniqueness and global estimates that are on par with those of the underlying homogeneous elliptic operator. Our results, in particular, establish the Green functions for Schrodinger, magnetic Schrodinger, and generalized Schrodinger operators with real or complex coefficients on arbitrary domains

The unitary perturbations of a given unitary operator by finite rank d operators can be parametrized by dxd unitary matrices; this generalizes the rank one setting, where the Clark family is parametrized by the scalars on the unit circle. For finite rank perturbations we investigate the functional model of a related class of contractions, as well as a (unitary) Clark operator that realizes such a model representation. We express the adjoint of the Clark operator through a matrix-valued Cauchy integral operator. We determine the matrix-valued characteristic functions of the model (for contractions). In the case of inner characteristic functions results suggest a generalization of the normalized Cauchy transform to the finite rank setting. This presentation is based on joint work with Sergei Treil

Let $m$ be a radial multiplier supported in a compact subset away from the origin. For dimensions $d\ge 2$, it is conjectured that the multiplier operator $T_m$ is bounded on $L^p(R^d)$ if and only if the kernel $K=\hat{m}$ is in $L^p(R^d)$, for the range $1<p<2d/(d+1)$. Note that there are no a priori assumptions on the regularity of the multiplier. This conjecture belongs near the top of the tree of a number of important related conjectures in harmonic analysis, including the Local Smoothing, Bochner-Riesz, Restriction, and Kakeya conjectures. We discuss new progress on this conjecture in dimensions $d=3$ and $d=4$. Our method of proof will rely on a geometric argument involving sizes of multiple intersections of three-dimensional annuli